Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in four and more spatial dimensions

Abstract

We study the Cauchy problem for the Klein-Gordon-Zakharov system in spatial dimension $d\ge 4$ with radial or non-radial initial datum $(u,{\partial }_{t}u,n,{\partial }_{t}n){|}_{t=0}\in H^{s+1}({\mathbb{R}}^d)\times H^s({\mathbb{R}}^d)\times {\dot{H}}^s({\mathbb{R}}^d)\times {\dot{H}}^{s-1}({\mathbb{R}}^d)$. The critical value of $s$ is $s=s_c=d/2-2$. If the initial datum is radial, then we prove the small data global well-posedness and scattering at the critical space in $d\ge 4$ by applying the radial Strichartz estimates and $U^2,V^2$ type spaces. On the other hand, if the initial datum is non-radial, then we prove the local well-posedness at $s=1/4$ when $d=4$ and $s=s_c+1/(d+1)$ when $d\ge 5$ by applying the $U^2,V^2$ type spaces.

References in corpus (2)

• Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data
• Global existence without decay for quadratic Klein-Gordon equations